6r^2=4r+17

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Solution for 6r^2=4r+17 equation:


Simplifying
6r2 = 4r + 17

Reorder the terms:
6r2 = 17 + 4r

Solving
6r2 = 17 + 4r

Solving for variable 'r'.

Reorder the terms:
-17 + -4r + 6r2 = 17 + 4r + -17 + -4r

Reorder the terms:
-17 + -4r + 6r2 = 17 + -17 + 4r + -4r

Combine like terms: 17 + -17 = 0
-17 + -4r + 6r2 = 0 + 4r + -4r
-17 + -4r + 6r2 = 4r + -4r

Combine like terms: 4r + -4r = 0
-17 + -4r + 6r2 = 0

Begin completing the square.  Divide all terms by
6 the coefficient of the squared term: 

Divide each side by '6'.
-2.833333333 + -0.6666666667r + r2 = 0

Move the constant term to the right:

Add '2.833333333' to each side of the equation.
-2.833333333 + -0.6666666667r + 2.833333333 + r2 = 0 + 2.833333333

Reorder the terms:
-2.833333333 + 2.833333333 + -0.6666666667r + r2 = 0 + 2.833333333

Combine like terms: -2.833333333 + 2.833333333 = 0.000000000
0.000000000 + -0.6666666667r + r2 = 0 + 2.833333333
-0.6666666667r + r2 = 0 + 2.833333333

Combine like terms: 0 + 2.833333333 = 2.833333333
-0.6666666667r + r2 = 2.833333333

The r term is -0.6666666667r.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667r + 0.1111111112 + r2 = 2.833333333 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667r + r2 = 2.833333333 + 0.1111111112

Combine like terms: 2.833333333 + 0.1111111112 = 2.9444444442
0.1111111112 + -0.6666666667r + r2 = 2.9444444442

Factor a perfect square on the left side:
(r + -0.3333333334)(r + -0.3333333334) = 2.9444444442

Calculate the square root of the right side: 1.715938357

Break this problem into two subproblems by setting 
(r + -0.3333333334) equal to 1.715938357 and -1.715938357.

Subproblem 1

r + -0.3333333334 = 1.715938357 Simplifying r + -0.3333333334 = 1.715938357 Reorder the terms: -0.3333333334 + r = 1.715938357 Solving -0.3333333334 + r = 1.715938357 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + r = 1.715938357 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + r = 1.715938357 + 0.3333333334 r = 1.715938357 + 0.3333333334 Combine like terms: 1.715938357 + 0.3333333334 = 2.0492716904 r = 2.0492716904 Simplifying r = 2.0492716904

Subproblem 2

r + -0.3333333334 = -1.715938357 Simplifying r + -0.3333333334 = -1.715938357 Reorder the terms: -0.3333333334 + r = -1.715938357 Solving -0.3333333334 + r = -1.715938357 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + r = -1.715938357 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + r = -1.715938357 + 0.3333333334 r = -1.715938357 + 0.3333333334 Combine like terms: -1.715938357 + 0.3333333334 = -1.3826050236 r = -1.3826050236 Simplifying r = -1.3826050236

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.0492716904, -1.3826050236}

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